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Derivation of the radiative transfer equation forscattering media with a spatially varying refractive index

Jean-Michel Tualle, Eric Tinet

To cite this version:Jean-Michel Tualle, Eric Tinet. Derivation of the radiative transfer equation for scattering mediawith a spatially varying refractive index. Optics Communications, Elsevier, 2003, 228, pp.33-38.hal-00002848

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Derivation of the radiative transfer equation

for scattering media with a spatially varying refractive index.

Jean-Michel Tualle and Eric Tinet

Laboratoire de Physique des Lasers (CNRS UMR 7538), Université Paris 13

99 av. J.-B. Clément, 93430 Villetaneuse, France

ABSTRACT:

We derive in this paper a radiative transfer equation in scattering media with spatially

varying refractive index, together with its associated diffusion approximation. We present an

approximate result of this diffusion equation in a simple case, and this result is compared to

Monte Carlo simulations.

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A radiative transfer equation (RTE) in scattering media with a spatially varying

refractive index has been derived by H. Ferwerda [1]; T. Khan and H. Jiang [2] deduced the

diffusion equation associated to this result. The equation of H. Ferwerda however does not

satisfy energy conservation, as can be seen in [2]. In this paper we will reconsider the

derivation of the RTE, and will obtain a completely different result. In order to prove the

pertinence of our result, we will deduce a diffusion equation from our RTE; we will then

obtain an analytical approximate result in the case of low refractive index gradient, for both

diffusion equations derived by Khan et al and us. These analytical results are then compared

to a Monte Carlo simulation.

The RTE comes from an energy balance in a small cylinder represented on figure 1.

Let us consider this energy balance at point 0r , with a direction of propagation parallel to 0Ω

(within the solid angle Ωd ). The energy enters to the cylinder at point 0r and at time t,

passing through the elementary area dA , and it leaves at point 'r and at time t + dt, through

'dA , after a travel ndtccdtds /0== (n is the varying refractive index). The surface elements

dA and dA’ are chosen to be orthogonal to the rays that are crossing them; 0Ω

is therefore

orthogonal to the entry face dA.

Let us recall the definition of the radiance ),,( 00 trL Ω , which is defined so that

uddAL ⋅ΩΩ 0 is the power flowing within the solid angle Ωd through dA (u is a unit

vector orthogonal to dA ). The energy flow can be materialized by a vector field ),( 00ΩΩ

rr ,

where Ω

is a unit vector parallel at each point r to the geometrical optical ray passing

through this point. The definition of the radiance is a local one, and this vector field has to be

considered only at the vicinity of 0r ; if r is a point on the entry face dA, we can set

00 ),(0

Ω=ΩΩ

rr , which means that, in accordance with the definition of L, we only consider

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here the rays that propagate towards the direction 0Ω

. If however r is a point on the ray

passing through 0r , at a distance ds from 0r

, we have to use the fundamental equation of

geometrical optics [1].

nPnds

dk

iki

∂=Ω0

1 (1)

where

[ ]kiikikP 000 ΩΩ−= δ (2)

is the projector on the plane orthogonal to 0Ω

and where we have used the Einstein’s

summation convention in order to simplify our expressions; we of course have in Euclidean

space ii aa = . Let us now find the matrix i

jΩ∂ that links at 0rr = an infinitesimal variation

idΩ to an infinitesimal displacement jdl . Such a matrix is of course unique, so that one only

have to check our solution for every jdl⊥ that are orthogonal to 0Ω

, and every jdl // that are

parallel to 0Ω

. This can be readily done for the following derivation rule:

[ ] nn k

kiikjij ∂ΩΩ−

Ω=Ω∂ 00

0 δ (3)

One indeed have 0=Ω∂ ⊥ji

j dl and dsds

ddsdli

jij

jij

Ω=ΩΩ∂=Ω∂ 0// . Let us now write the

energy balance in the cylinder of figure 1, which reads [3]:

dAddstr

dAddtrLfds

dAdtrLds

dAdtrLdAddttrL

s

sa

ΩΩ+

ΩΩ+

ΩΩ+−=

ΩΩ−Ω+Ω

∫),,(

),,(),(

),,()(

),,(''),','(

00

400

00

00

ε

ωωωµ

µµ

π

(4)

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where aµ is the absorption coefficient per unit length and sµ is the scattering coefficient per

unit length. ),( 0 ω

Ωf is the phase function, which gives the probability for an energy packet

traveling in direction ω to be scattered into direction 0Ω

. f is normalized according to

1),(4

0 =Ω∫ ωωπ

df . To finish with, ε is a source distribution per unit volume and unit solid

angle. In order to calculate 'dA , we first notice with H. Ferwerda [1] that we have up to the

first order in ds , dA :

dAdsdAdArd ii

cyl

ii Ω∂=−=Ω∂∫ '3 (5)

We get however from (3) 0=Ω∂ ii , so that dAdA =' . This point leads to a first difference

with the result of H. Ferwerda [1], who has:

031≠

∂Ω−∂Ω

=Ω∂ ∑ nn

nn i

iii

i

ii

This expression is however not covariant, as the left member is a scalar and the right

one is not ( the term iΩ1 does not transform as a vector so that i

ii n Ω∂∑ / is not a scalar):

this equation is therefore not preserved through a rotation of the Descartes frame, what leads

to difficulties for its physical interpretation. Another difference with the result of H. Ferwerda

comes from the solid angle 'Ωd we take here explicitly into account. Let us introduce 2

vectors )1(0yd and )2(

0yd orthogonal to 0Ω

and define the solid angle Ωd as:

),,det()( )2(0

)1(00

)2(0

)1(00 ydydydydd

Ω=×⋅Ω=Ω (6)

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If )(00

)(0

NN yd

+Ω=Ω , we can define the vector field ),( )(0

)(0

Nr

N r ΩΩ=Ω

. We want the

variation of )(NΩ

for an infinitesimal displacement along )(0NΩ

, that is:

),( )(00

)(0

)(

0

Nr

NN

rds

dΩΩ∇⋅Ω=Ω

dsd N )(Ω

is a function of )(0NΩ

that we can develop around 0Ω

:

dsddy

dsd

dsd

k

kNN Ω

∂+Ω=ΩΩ

0

)(0

)(

(7)

We can furthermore write:

)()(0

)(NN

N

yddsd

dsd

+Ω∇⋅Ω=Ω

As we have already seen that 0)(0 =Ω∇⋅

Nyd , so that dsdN Ω=Ω∇⋅Ω=Ω∇⋅Ω

0)(

0 , we can

therefore conclude:

dsddydy

dsd i

kNiNk

Ω∂= Ω0

)(0

)( (8)

and

Ω∂

Ω∂

Ω+Ω=Ω ΩΩ ),,(),,(det' )2(0

)1(0

)2(0

)1(00 00

ki

kii

iii dyds

ddyds

dds

ddsdydydkk

(9)

We will first notice that dsd /Ω

, which is orthogonal to 0Ω

, is in the plane ),( )2(0

)1(0 ydyd and

do not contribute in (9) up to the first order in ds; a second remark is that:

),,0(),,( )2(0

)1(0

)2(0

)1(00

iijjjij dydydydyP =Ω (10)

which allows us to write:

ΩΩ

∂+=Ω

Ω∂+=Ω ΩΩ dP

dsddsdP

dsddsd k

i

ik

l

iil kk

)1(det'00

δ (11)

We finally obtain:

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Ω∇⋅Ω−=Ω dnn

dsd )21(' 0

(12)

The derivation of the RTE presents no other difficulties, and following H. Ferwerda [1] we

obtain:

εωωωµµµπ

+Ω++−=

∇⋅∇+∇⋅Ω−∇⋅Ω+∂∂

∫

Ω

dLfL

Lnn

Lnn

LtL

c

ssa )(),()(

1)(21

40

00 0

(13)

Where we have used, as noticed in [2], 000 =∇⋅Ω Ω L

. In the following we will drop

the subscript 0 and use Ω

instead of 0Ω

in (13). If we integrate now (13) with respect to Ω

over 4π srad, if we introduce:

- the average diffuse intensity ∫ ΩΩ=π

ϕ4

),,(),( dtrLtr

- the diffuse flux vector ∫ ΩΩΩ=π4

),,(),( dtrLtrj

- the source term ∫ ΩΩ=π

ε4

),,(),( dtrtrE

and if we notice with Khan et al [2] that jdL

24

=Ω∇∫ Ωπ

, we directly obtain from (13) the

conservation equation:

Ejtc a =+⋅∇+∂∂ ϕµϕ 1 (14)

This equation warrant the conservation of the energy density W=ϕ / c, which was not

the case with the equation of H. Ferwerda (see [2]) and is a first point that validate the RTE

derived in this paper. Let us now derive a diffusion equation from (13), in order to perform

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some comparisons with Monte Carlo simulations. The diffusion approximation consists in

setting [2,3]:

Ω⋅+=Ω ),(

43),(

41),,( trjtrtrI

πϕ

π(15)

Let us insert (15) in (13), multiply by Ω

and integrate with respect to Ω

. An integral

of a product of an odd number of Ω

components, like ∫ ΩΩπ4

d

or ∫ ΩΩΩΩπ4

dkji , is null for

obvious symmetry arguments. We furthermore have the well-known relations [3]:

ijji d δπ

π 34

4

=ΩΩΩ∫

and

ijji gddf δωωωπ π 3

),(41

4

=ΩΩΩ∫

where ωωωπ π

ddfg Ω⋅ΩΩ= ∫

4

),(41 is the anisotropy factor. From these relations we obtain:

[ ] Fjgnnt

jc sa +−+−=∇−∇+∂∂

µµϕϕ )1(32

311

where ∫ ΩΩΩ=π

ε4

),,(),( dtrtrF . Assuming 0

=

∂∂tj leads to the modified Fick law:

FDnnDDj

32 +∇+∇−= ϕϕ (16)

The diffusion equation comes from the insertion of (16) in the conservation equation (14):

[ ] [ ]FDEnnDD

tc a

⋅∇−=+

∇⋅∇+∇⋅∇−

∂∂ 321 ϕµϕϕϕ (17)

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which has to be compared to the diffusion equation obtained by Khan et al [2] from the result

of H. Ferwerda [1]:

[ ] [ ]FDFnnDEn

nDD

tc a

⋅∇−⋅∇−=+∇⋅∇−∇⋅∇−

∂∂ 3621 ϕµϕϕϕ (18)

If we consider now the simple case with 0=aµ , D constant, rqnn ⋅+= 0 with

constant q , and with an isotropic point-like source ( )()( trE δδ = , 0

=F ), we obtain for

equation (17):

)()(221 22 trq

nDq

nDD

tcδδϕϕϕϕ =−∇⋅+∆−

∂∂ (19)

and for equation (18):

)()(21 trqnDD

tcδδϕϕϕ =∇⋅−∆−

∂∂ (20)

The leading difference between (19) and (20) is therefore the sign of the term ϕ∇⋅q .

Let us search a solution of (19) of the form:

+−= − y

ctDrtA

4exp

22

3ϕ (21)

If we assume that y is of the same order than q, and if we neglect the second order

terms, inserting (21) in (19) for t>0 leads to:

0)(4

12

2

=⋅−

∇⋅+∆−∂∂

ctnDqrr

ctyryD

ty

c

(22)

and we find:

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rqctD

rn

y ⋅

+=

25

421 2

(23)

The coefficient A in (21) is obtained by comparison, in the limit 0→t , to the

classical result without variation of the refractive index [4]:

2/300

0

0 )/4( −= nDcnc

A π (24)

Concerning equation (20) the substitution (21) leads to:

0)(4

212

2

=⋅−

⋅+∇⋅+∆−∂∂

ctnDqrr

nctqr

ctyryD

ty

c

(25)

that is:

rqctD

rn

y ⋅

−=

23

421 2

(26)

with of course the same value for A.

Let us now compare these results with Monte Carlo simulations [5]. We stress here on

the fact that Monte Carlo simulations are based on the propagation of multiple random

walkers that propagate on optical rays and experience absorption and scattering events: These

simulations are therefore completely disconnected from the RTE, and can be used to

discriminate all the results presented here. We simulate a scattering medium with 0=aµ ,

150 −= cmsµ , an Henyey-Greenstein phase function with an anisotropy factor g = 0.8, and a

refractive index zqnn += 0 with 5.10 =n and 102.0 −= cmq . The average diffuse intensity

obtained from these simulations is presented in figure 2 for a point located on the z axis at

cmz 2= . The noisy curve comes from the Monte Carlo simulations, and the bold line

corresponds to our result (equations (21,23,24) ). The result obtained from the work of H.

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Ferwerda (equations (21,24,26) ) is presented with dashed line, and the dotted line in the

middle of the three curves is the average diffuse intensity ϕ0 without refractive index gradient,

that is with 0=q . As can be seen in this figure, there is a fundamental difference between our

result and the result of H. Ferwerda, as H. Ferwerda predicts a decrease of the average diffuse

intensity when we predict an increase of this quantity, in accordance with Monte Carlo

simulations. This point is more striking in figure 3, where we plot the quantities (ϕ-ϕ0)/ϕ0

corresponding to the curves of figure 2. In fact the equation of geometrical optics (1) predicts

that the rays are deviated toward the direction of increasing refractive index values, so one

should intuitively await an increase of the diffuse intensity in the direction of the refractive

index gradient. Our result presents a very satisfying correspondence with Monte Carlo

simulation. We recall that it is obtained without any adjustable parameter. We present in

figure 4 and 5 the same results for cmz 2−= , with the same conclusions.

As a conclusion, we derived a radiative transfer equation in scattering media with

spatially varying refractive index, together with a diffusion approximation solved in a simple

case. Our results are in very good correspondence with Monte Carlo simulations.

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References

1 Ferwerda H. 1999 “The radiative transfer equation for scattering media with a

spatially varying refractive index”, J. Opt A.: Pure Appl. Opt. 1, L1-L2.

2 Khan T. and Jiang H. 2003 “A new diffusion approximation to the radiative

transfer equation for scattering media with spatially varying refractive indices”, J.

Opt A.: Pure Appl. Opt. 5, 137-141.

3 Ishimaru A. 1978 Wave Propagation and Scattering in Random Media (New

York: Academic).

4 M.S. Patterson, B. Chance and B.C Wilson 1989 “Time resolved reflectance and

transmittance for the non invasive measurement of tissue optical properties”,

Applied Optics 28 , 2331-2336.

5 E. Tinet 1992, PhD thesis (in French), Université Paris 13.

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Figure Captions

Figure 1:

Cylinder on which the energy balance is considered. The vector field Ω

is parallel at

each point r to the geometrical optical ray passing through this point.

Figure 2:

Average diffuse intensity for cmz 2= . The noisy curve comes from the Monte Carlo

simulations and the bold line corresponds to our result. The result obtained from the work of

H. Ferwerda is presented with dashed line, and the dotted line in the middle of the three

curves corresponds to the case 0=q .

Figure 3:

(ϕ-ϕ0)/ ϕ0 for cmz 2= . The noisy curve comes from the Monte Carlo simulations and

the bold line corresponds to our result. The result obtained from the work of H. Ferwerda is

presented with dashed line.

Figure 4:

Average diffuse intensity for cmz 2−= . The noisy curve comes from the Monte Carlo

simulations, and the bold line corresponds to our result. The result obtained from the work of

H. Ferwerda is presented with dashed line, and the dotted line in the middle of the three

curves corresponds to the case 0=q .

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Figure 5:

(ϕ-ϕ0)/ ϕ0 for cmz 2−= . The noisy curve comes from the Monte Carlo simulations

and the bold line corresponds to our result. The result obtained from the work of H. Ferwerda

is presented with dashed line.

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Ω

r

'r 'Ω

0r

0Ω

dA’

dA

Figure 1

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0 1000 2000 3000 4000 50000,00000

0,00005

0,00010

0,00015

0,00020

ϕ(c

m-2

ps-1

)

t (ps)

Figure 2

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0 1000 2000 3000 4000 5000

-4

-2

0

2

4

(ϕ-ϕ

0)/ϕ 0

(in

%)

t (ps)

Figure 3

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0 1000 2000 3000 4000 5000 6000

0,00000

0,00005

0,00010

0,00015

0,00020

ϕ(c

m-2

ps-1

)

t (ps)

Figure 4

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0 1000 2000 3000 4000 5000

-4

-2

0

2

4

(ϕ-ϕ

0)/ϕ 0

(in

%)

t (ps)

Figure 5

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